\section{Preference Game Definition} \label{sec:prefGameDef}
We next define a new, simple game, the preference game, which can be reduced to both FBBC and FSPP games. We use preference games to show \PPAD-hardness of finding FBBC and FSPP equilibria.  In a preference game with a set $S$ of
players, each player's strategy set is $S$.  Each player $i \in S$ has
a preference relation $\geqprefer_i$ among the strategies (a partial order). Each player
$i$ chooses a {\em weight distribution}, which is an assignment $w_i: S
\rightarrow [0,1]$ satisfying two conditions: (a) the weights add up
to $1$: $\sum_{j \in S} w_i(j) = 1$; and (b) the weight placed by $i$
on $j$ is no more than the weight placed by $j$ on $j$: $w_i(j) \le
w_j(j)$ for all $i,j \in S$.  As in the case of FSPP, the
preference relations $\geqprefer_i$ induce a preference relation among
the weight distributions as follows: $w_i$ is {\em lexicographically
at least}\/ $w'_i$ if for all $j \in S$, $\sum_{k \geqprefer_i j} w_i(k)
\ge \sum_{k \geqprefer_i j} w'_i(k)$.  An equilibrium in a preference game is an
assignment $w = \{w_i: i \in S\}$ such that $w_i$ is lexicographically
maximal for all $i \in S$.  

Preference games show up as a simplification of a number of real-world situations, aside from FBBC and FSPP games.  
\begin{itemize}
\item \BfPara{Party Planning} On New Year's Eve, you and each of your friends are considering hosting a party. Each of
you has a preference order over the others' parties and has to
determine the fraction of the evening that you will spend at each
party.  Naturally, one cannot spend more time at a party
than the person hosting that particular party.  Your optimal action --
how long to host your party and which other parties to attend for how
long -- depends on your preference and other players' actions. 
\item \BfPara{Condo Association Meetings} Your condo association gives you the option of attending the meetings or else designating another member of the association who will get to vote on your behalf on all issues that arise. You may choose to attend only a fraction of the meetings. You trust the opinions of some members more than others to vote according to your wishes, giving a preference order across members. Somewhere in the order is the option of spending the time to attend the meetings (or part of the meetings) yourself. You cannot have another member take your vote more often than the other member attends. Your optimal action is the fraction of the meetings you will attend and the fraction of time each other member will take your vote.
\item \BfPara{Blog Content} Bloggers may want to fill their blogs with a combination of original content and content copied from other blogs, with a preference for which blogs are copied. Of course, more cannot be copied from another blog than that other blogger has written, and we can assume that bloggers only copy content directly from its original source (i.e., they never copy from a copy). The preference game models each blogger's choice of what percentage of his blog is original and what percentages are copied from which other bloggers. \footnote{Thanks to David Karger for suggesting the blog content motivation for the preference game.}
\item \BfPara{Facility Location} The preference game can be viewed as a simple, preference-based model of facility location in which the clients themselves make decisions about the building locations. Each client may build its own, local facility and produce the commodity locally or may choose to use a facility built at another client site. The client has a preference order across locations and must account for 100\% of his needs. He may produce only a fraction of his needs locally, or may use another facility for only a fraction of his needs, but he may not use another client's facility for a larger percentage than the amount of facility that has been built by that client.
\end{itemize}
